## Thursday, August 4, 2011

### Measuring Investment Performance with Arithmetic and Geometric Averages

PERSONAL FINANCE 101

Say you're choosing between two fund managers, Tyrion and Bronn, both of whom have only been in business for two years. In choosing between the two, you decide to use their actual performance in the past two years as an indicator of their "investment skill," thinking that how well they did in the past would be a good sign of how well they'll do in the future (some experts might disagree with this). The historical returns of the portfolios managed by Tyrion and Bronn are as follows:

 % Returns Year 1 Year 2 Tyrion -50 100 Bronn 10 20

Given this information, who would you entrust your money with, Tyrion or Bronn?

Most of us would probably to it this way: get the average of the returns of the two managers, and choose the manager with the higher average return. For Tyrion, it would be (-50% + 100%)/2 = 25% per year; for Bronn, it's (20% + 20%)/2 = 15% per year. Since Tyrion has earned a higher average return than Bronn, then you should choose him to be your fund manager. Makes sense, right?

No, it actually doesn't! What's wrong with this picture? I'll give you a few moments to figure it out...

Bronn and Tyrion: Happier times

Here's what's wrong with the previous analysis: Tyrion hasn't actually performed better than Bronn in the past two years, regardless of what our computations tell us. In fact, if you invested your money with Tyrion two years ago, you would have ended up with the same amount today, compared to if you invested with Bronn where you would have earned 32% more than what you started with over two years.

The problem is that we used simple or arithmetic average to compute for the average annual return of the two fund managers; based on what we saw above, it's clear that this method is inadequate for such a purpose. What's more appropriate to use for this kind of analysis would be the geometric average; the geometric average of n returns is given by

geometric average return = [(1 + r1)(1 + r2)...(1 + rn)]^(1/n) - 1

Using the example above, we get the annual geometric return for Tyrion and Bronn:

Tyrion: [(1 - 0.5)(1 + 1)]^(1/2) - 1 = 0% per year
Bronn: [(1 + 0.1)(1 + 0.2)]^(1/2) -1 = 14.9% per year

Which is consistent with our earlier observation that Bronn actually outperformed Tyrion in the past two years, and specifically, that you would have ended back where you started (i.e., zero return) had you entrusted your money with Tyrion.

That's it. Just remember, in assessing past performance of managers, funds, or portfolios, always use  geometric average instead of arithmetic average.